In practice, the error variance \(\sigma ^2\)  in the regression model is unknown and needs to be estimated. Let \(\sigma ^2_0\)  be its true value. [1]} proposed a fully Bayesian scheme by endowing the standard error \(\sigma \)  with a hyperprior, which is supported on a compact interval \([a,b]\subset (0,\infty )\)  that contains \(\sigma _0\)  with a Lebesgue density bounded away from zero. This approach has been followed by many others [2]}, [3]}. [4]} showed a Bernstein-von Mises theorem for the marginal posterior of \(\sigma \) , where the prior for \(\sigma \)  is relaxed to be supported on \((0,\infty )\) . In other words, it is possible to simultaneously estimate the regression function at an optimal nonparametric rate and the standard deviation at a parametric rate.
